A new theoretical framework for the formulation of general, nonlinear, multiscale plate theories

Abstract

A new type of general, theoretical framework for the development of comprehensive, nonlinear, multiscale plate theories for laminated structures is presented. The theoretical framework utilizes a generalized two scale description of the displacement field based on a superposition of global and local effects where the functional forms for the global and local displacement fields are arbitrary. The two scale nature of theory allows it to explicitly consider the layered nature of the structure. The subsequent development of the governing equations for the theory is carried out using the general nonlinear equations of continuum mechanics referenced to the initial configuration. The equations of motion and the lateral surface boundary conditions for the theory are derived using the method of moments over the different scales subject to an orthogonality constraint. The theory satisfies the interfacial constraints and the top and bottom surface boundary conditions in a strong sense. Delamination effects are incorporated into the theory through the use of cohesive zone models (CZMs). Arbitrary CZMs can be incorporated into the theory without the need for reformulation of the governing equations. The theory is formulated in a sufficiently general fashion that any type of history-dependent material can be used to describe the inelastic response of the materials composing the layers. Furthermore, as a result of the multiscale nature of the theory it can be specialized to single scale theories of the equivalent single layer (ESL) or discrete layer (DL) types in a unified fashion and without the need for any reformulation. While the starting point for the proposed theory is the same as used by Williams [Williams, T.O., 1999. A generalized multilength scale nonlinear composite plate theory with delamination. Int. J. Solid Struct. 36, (20) 3015–3050; Williams, T.O., 2001. Efficiency and accuracy considerations in a unified plate theory with delaminations. Comp. Struct. 52, (1) 27–40; Williams, T.O., 2005. A generalized, multilength scale framework for thermo-diffusionally-mechanically coupled, nonlinear, laminated plate theories with delaminations. IJSS 42, (5–6) 1465–1490] the subsequent formulation is significantly different. The differences in the two theories allow the currently proposed theory to improve on the capabilities of the previous theory; particularly in the satisfaction of the traction continuity constraints at the interfaces. It is shown that the theory is capable of providing accurate predictions for all of the fields in perfectly bonded and delaminated plates even for relatively low orders of displacement approximations. In particular, the theory is shown to provide accurate predictions for the transverse stresses that are continuous across the interfaces directly from the constitutive relations.